L(s) = 1 | + (−0.959 − 0.281i)2-s + (1.25 − 1.45i)3-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (−1.61 + 1.03i)6-s + (0.345 + 0.755i)7-s + (−0.654 − 0.755i)8-s + (−0.381 − 2.65i)9-s + (0.415 − 0.909i)10-s + (1.84 − 0.540i)12-s + (−0.118 − 0.822i)14-s + (1.25 + 1.45i)15-s + (0.415 + 0.909i)16-s + (−0.381 + 2.65i)18-s + (−0.654 + 0.755i)20-s + (1.52 + 0.449i)21-s + ⋯ |
L(s) = 1 | + (−0.959 − 0.281i)2-s + (1.25 − 1.45i)3-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (−1.61 + 1.03i)6-s + (0.345 + 0.755i)7-s + (−0.654 − 0.755i)8-s + (−0.381 − 2.65i)9-s + (0.415 − 0.909i)10-s + (1.84 − 0.540i)12-s + (−0.118 − 0.822i)14-s + (1.25 + 1.45i)15-s + (0.415 + 0.909i)16-s + (−0.381 + 2.65i)18-s + (−0.654 + 0.755i)20-s + (1.52 + 0.449i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8281799683\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8281799683\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (0.959 + 0.281i)T \) |
good | 3 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 7 | \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 17 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 19 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 29 | \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \) |
| 31 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 41 | \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \) |
| 43 | \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + 0.284T + T^{2} \) |
| 53 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 59 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 61 | \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \) |
| 67 | \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 71 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 79 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 83 | \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 97 | \( 1 + (0.959 + 0.281i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.29868146232255924473062359527, −9.993010713726863114378531035350, −9.159501505886341486083140523919, −8.188029410847674472327900817327, −7.82889138959312838208494163486, −6.80005337923226235243679443240, −6.18339278268621029179183478186, −3.52415500206230296535913446515, −2.61989487524056147674429014415, −1.78484961541532501920619003560,
1.97729578106915619157977245097, 3.57263244467458726209747405409, 4.56350235431036066847800588024, 5.55989719576445126113024418349, 7.39156779526518463546868810782, 8.136987451593417341837442212695, 8.755556068806536079785110445606, 9.522710747798696199751936780274, 10.19537309865978822326717032335, 10.91483425537526987423312165401